Question

The pair of linear equations $x + y = 3,2x + 5y = 12$ has a unique solution $x = {x_1},y = {y_1}$ then the value of ${x_1}$ is?
A ) 1
B ) 2
C ) $ - 1$
D ) $ - 2$

Answer 1

Hint:- We will proceed by using the graphical analysis of intersection of two linear equations. Since there is a unique point which satisfies the two equations, they should intersect at one point.

Complete step-by-step answer:
According to the question, we know that the given equations are
$x + y = 3$
$2x + 5y = 12$
Also given is that the system of these two equations has a unique solution at $x = {x_1}$and $y = {y_1}$
From this statement, we can infer that the lines represented by the two given equations intersect at only one point which has the coordinate $\left( {{x_1},{y_1}} \right)$
Thus from graphical analysis, we can conclude that this point lies on both the lines, that is, it should satisfy both the equations.
By putting $x = {x_1}$and $y = {y_1}$in the given equations, we get
${x_1} + {y_1} = 3...\left( 1 \right)$
$2{x_1} + 5{y_1} = 12....\left( 2 \right)$
Rearranging equation $\left( 1 \right)$ we can write ${y_1}$ in terms of ${x_1}$
Thus, we get ${y_1} = 3 - {x_1}$
Putting the values of ${y_1}$ in equation $\left( 2 \right)$ , we get
$2{x_1} + 5\left( {3 - {x_1}} \right) = 12$
By using simplification and rearrangement, we can solve the above equation to get the value of ${x_1}$
$ \Rightarrow 2{x_1} + 15 - 5{x_1} = 12$
$ \Rightarrow 2{x_1} - 5{x_1} = 12 - 15$
$ \Rightarrow - 3{x_1} = - 3$
$ \Rightarrow {x_1} = 1$

Hence, the correct answer is (A)

Note:- In these types of questions, it is important to analyse whether the given pair of linear equations, a system of equations in this case, have a unique solution, no solution or infinite solution. In case of a unique solution, there exists only one point which satisfies both the equations, that is which represents a single point of intersection from the graphical analysis of view. This unique point satisfies the equations algebraically.